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Journal Cover Physica D: Nonlinear Phenomena
  [SJR: 1.048]   [H-I: 89]   [3 followers]  Follow
    
   Hybrid Journal Hybrid journal (It can contain Open Access articles)
   ISSN (Print) 0167-2789
   Published by Elsevier Homepage  [2970 journals]
  • Fractional Schrödinger dynamics and decoherence
    • Abstract: Publication date: Available online 16 June 2016
      Source:Physica D: Nonlinear Phenomena
      Author(s): Kay Kirkpatrick, Yanzhi Zhang
      We study the dynamics of the Schrödinger equation with a fractional Laplacian ( − Δ ) α , and the decoherence of the solution is observed. Analytically, we obtain equations of motion for the expected position and momentum in the fractional Schödinger equation, equations that are the fractional counterpart of the well-known Newtonian equations of motion for the standard ( α = 1 ) Schrödinger equation. Numerically, we propose an explicit, effective numerical method for solving the time-dependent fractional nonlinear Schrödinger equation–a method that has high order spatial accuracy, requires little memory, and has low computational cost. We apply our method to study the dynamics of fractional Schrödinger equation and find that the nonlocal interactions from the fractional Laplacian introduce decoherence into the solution. The local nonlinear interactions can however reduce or delay the emergence of decoherence. Moreover, we find that the solution of the standard NLS behaves more like a particle, but the solution of the fractional NLS behaves more like a wave with interference effects.


      PubDate: 2016-06-18T18:04:02Z
       
  • Entropy rates of low-significance bits sampled from chaotic physical
           systems
    • Abstract: Publication date: Available online 16 June 2016
      Source:Physica D: Nonlinear Phenomena
      Author(s): Ned J. Corron, Roy M. Cooper, Jonathan N. Blakely
      We examine the entropy of low-significance bits in analog-to-digital measurements of chaotic dynamical systems. We find the partition of measurement space corresponding to low-significance bits has a corrugated structure. Using simulated measurements of a map and experimental data from a circuit, we identify two consequences of this corrugated partition. First, entropy rates for sequences of low-significance bits more closely approach the metric entropy of the chaotic system, because the corrugated partition better approximates a generating partition. Second, accurate estimation of the entropy rate using low-significance bits requires long block lengths as the corrugated partition introduces more long-term correlation, and using only short block lengths overestimates the entropy rate. This second phenomenon may explain recent reports of experimental systems producing binary sequences that pass statistical tests of randomness at rates that may be significantly beyond the metric entropy rate of the physical source.


      PubDate: 2016-06-18T18:04:02Z
       
  • Nonlinear wave dynamics near phase transition in PT-symmetric localized
           potentials
    • Abstract: Publication date: 15 September 2016
      Source:Physica D: Nonlinear Phenomena, Volume 331
      Author(s): Sean Nixon, Jianke Yang
      Nonlinear wave propagation in parity-time symmetric localized potentials is investigated analytically near a phase-transition point where a pair of real eigenvalues of the potential coalesce and bifurcate into the complex plane. Necessary conditions for a phase transition to occur are derived based on a generalization of the Krein signature. Using the multi-scale perturbation analysis, a reduced nonlinear ordinary differential equation (ODE) is derived for the amplitude of localized solutions near phase transition. Above the phase transition, this ODE predicts a family of stable solitons not bifurcating from linear (infinitesimal) modes under a certain sign of nonlinearity. In addition, it predicts periodically-oscillating nonlinear modes away from solitons. Under the opposite sign of nonlinearity, it predicts unbounded growth of solutions. Below the phase transition, solution dynamics is predicted as well. All analytical results are compared to direct computations of the full system and good agreement is observed.


      PubDate: 2016-06-18T18:04:02Z
       
  • Degenerate Bogdanov–Takens bifurcations in a one-dimensional
           transport model of a fusion plasma
    • Abstract: Publication date: 15 September 2016
      Source:Physica D: Nonlinear Phenomena, Volume 331
      Author(s): H.J. de Blank, Yu.A. Kuznetsov, M.J. Pekkér, D.W.M. Veldman
      Experiments in tokamaks (nuclear fusion reactors) have shown two modes of operation: L-mode and H-mode. Transitions between these two modes have been observed in three types: sharp, smooth and oscillatory. The same modes of operation and transitions between them have been observed in simplified transport models of the fusion plasma in one spatial dimension. We study the dynamics in such a one-dimensional transport model by numerical continuation techniques. To this end the MATLAB package cl_matcontL was extended with the continuation of (codimension-2) Bogdanov–Takens bifurcations in three parameters using subspace reduction techniques. During the continuation of (codimension-2) Bogdanov–Takens bifurcations in 3 parameters, generically degenerate Bogdanov–Takens bifurcations of codimension-3 are detected. However, when these techniques are applied to the transport model, we detect a degenerate Bogdanov–Takens bifurcation of codimension 4. The nearby 1- and 2-parameter slices are in agreement with the presence of this codimension-4 degenerate Bogdanov–Takens bifurcation, and all three types of L–H transitions can be recognized in these slices. The same codimension-4 situation is observed under variation of the additional parameters in the model, and under some modifications of the model.


      PubDate: 2016-06-18T18:04:02Z
       
  • Burgers equation with no-flux boundary conditions and its application for
           complete fluid separation
    • Abstract: Publication date: 15 September 2016
      Source:Physica D: Nonlinear Phenomena, Volume 331
      Author(s): Shinya Watanabe, Sohei Matsumoto, Tomohiro Higurashi, Naoki Ono
      Burgers equation in a one-dimensional bounded domain with no-flux boundary conditions at both ends is proven to be exactly solvable. Cole–Hopf transformation converts not only the governing equation to the heat equation with an extra damping but also the nonlinear mixed boundary conditions to Dirichlet boundary conditions. The average of the solution v ¯ is conserved. Consequently, from an arbitrary initial condition, solutions converge to the equilibrium solution which is unique for the given v ¯ . The problem arises naturally as a continuum limit of a network of certain micro-devices. Each micro-device imperfectly separates a target fluid component from a mixture of more than one component, and its input–output concentration relationships are modeled by a pair of quadratic maps. The solvability of the initial boundary value problem is used to demonstrate that such a network acts as an ideal macro-separator, separating out the target component almost completely. Another network is also proposed which leads to a modified Burgers equation with a nonlinear diffusion coefficient.


      PubDate: 2016-06-15T08:40:59Z
       
  • Spike-adding in parabolic bursters: The role of folded-saddle canards
    • Abstract: Publication date: Available online 31 May 2016
      Source:Physica D: Nonlinear Phenomena
      Author(s): Mathieu Desroches, Martin Krupa, Serafim Rodrigues
      The present work develops a new approach to studying parabolic bursting, and also proposes a novel four-dimensional canonical and polynomial-based parabolic burster. In addition to this new polynomial system, we also consider the conductance-based model of the Aplysia R15 neuron known as the Plant model, and a reduction of this prototypical biophysical parabolic burster to three variables, including one phase variable, namely the Baer-Rinzel-Carillo (BRC) phase model. Revisiting these models from the perspective of slow-fast dynamics reveals that the number of spikes per burst may vary upon parameter changes, however the spike-adding process occurs in an explosive fashion that involves special solutions called canards. This spike-adding canard explosion phenomenon is analysed by using tools from geometric singular perturbation theory in tandem with numerical bifurcation techniques. We find that the bifurcation structure persists across all considered systems, that is, spikes within the burst are incremented via the crossing of an excitability threshold given by a particular type of canard orbit, namely the true canard of a folded-saddle singularity. However there can be a difference in the spike-adding transitions in parameter space from one case to another, according to whether the process is continuous or discontinuous, which depends upon the geometry of the folded-saddle canard. Using these findings, we construct a new polynomial approximation of the Plant model, which retains all the key elements for parabolic bursting, including the spike-adding transitions mediated by folded-saddle canards. Finally, we briefly investigate the presence of spike-adding via canards in planar phase models of parabolic bursting, namely the theta model by Ermentrout and Kopell.


      PubDate: 2016-06-15T08:40:59Z
       
  • Dynamics of transcription-translation networks
    • Abstract: Publication date: Available online 2 June 2016
      Source:Physica D: Nonlinear Phenomena
      Author(s): D. Hudson, R. Edwards
      A theory for qualitative models of gene regulatory networks has been developed over several decades, generally considering transcription factors to regulate directly the expression of other transcription factors, without any intermediate variables. Here we explore a class of models that explicitly includes both transcription and translation, keeping track of both mRNA and protein concentrations. We mainly deal with transcription regulation functions that are steep sigmoids or step functions, as is often done in protein-only models, though translation is governed by a linear term. We extend many aspects of the protein-only theory to this new context, including properties of fixed points, description of trajectories by mappings between switching points, qualitative analysis via a state-transition diagram, and a result on periodic orbits for negative feedback loops. We find that while singular behaviour in switching domains is largely avoided, non-uniqueness of solutions can still occur in the step-function limit.


      PubDate: 2016-06-15T08:40:59Z
       
  • Combustion waves in hydraulically resistant porous media in a special
           parameter regime
    • Abstract: Publication date: Available online 7 June 2016
      Source:Physica D: Nonlinear Phenomena
      Author(s): Anna Ghazaryan, Stéphane Lafortune, Peter McLarnan
      In this paper we study the stability of fronts in a reduction of a well-known PDE system that is used to model the combustion in hydraulically resistant porous media. More precisely, we consider the original PDE system under the assumption that one of the parameters of the model, the Lewis number, is chosen in a specific way and with initial conditions of a specific form. For a class of initial conditions, then the number of unknown functions is reduced from three to two. For the reduced system, the existence of combustion fronts follows from the existence results for the original system. The stability of these fronts is studied here by a combination of energy estimates and numerical Evans function computations and nonlinear analysis when applicable. We then lift the restriction on the initial conditions and show that the stability results obtained for the reduced system extend to the fronts in the full system considered for that specific value of the Lewis number. The fronts that we investigate are proved to be either absolutely unstable or convectively unstable on the nonlinear level.


      PubDate: 2016-06-15T08:40:59Z
       
  • Mixed-Mode Oscillations in a piecewise linear system with multiple time
           scale coupling
    • Abstract: Publication date: Available online 10 June 2016
      Source:Physica D: Nonlinear Phenomena
      Author(s): S. Fernández-García, M. Krupa, F. Clément
      In this work, we analyze a four dimensional slow-fast piecewise linear system with three time scales presenting Mixed-Mode Oscillations. The system possesses an attractive limit cycle along which oscillations of three different amplitudes and frequencies can appear, namely, small oscillations, pulses (medium amplitude) and one surge (largest amplitude). In addition to proving the existence and attractiveness of the limit cycle, we focus our attention on the canard phenomena underlying the changes in the number of small oscillations and pulses. We analyze locally the existence of secondary canards leading to the addition or subtraction of one small oscillation and describe how this change is globally compensated for or not with the addition or subtraction of one pulse.


      PubDate: 2016-06-15T08:40:59Z
       
  • A principle of similarity for nonlinear vibration absorbers
    • Abstract: Publication date: Available online 11 June 2016
      Source:Physica D: Nonlinear Phenomena
      Author(s): G. Habib, G. Kerschen
      This paper develops a principle of similarity for the design of a nonlinear absorber, the nonlinear tuned vibration absorber (NLTVA), attached to a nonlinear primary system. Specifically, for effective vibration mitigation, we show that the NLTVA should feature a nonlinearity possessing the same mathematical form as that of the primary system. A compact analytical formula for the nonlinear coefficient of the absorber is then derived. The formula, valid for any polynomial nonlinearity in the primary system, is found to depend only on the mass ratio and on the nonlinear coefficient of the primary system. When the primary system comprises several polynomial nonlinearities, we demonstrate that the NLTVA obeys a principle of additivity, i.e., each nonlinear coefficient can be calculated independently of the other nonlinear coefficients using the proposed formula.


      PubDate: 2016-06-15T08:40:59Z
       
  • Resonance Van Hove singularities in wave kinetics
    • Abstract: Publication date: Available online 14 June 2016
      Source:Physica D: Nonlinear Phenomena
      Author(s): Yi-Kang Shi, Gregory L. Eyink
      Wave kinetic theory has been developed to describe the statistical dynamics of weakly nonlinear, dispersive waves. However, we show that systems which are generally dispersive can have resonant sets of wave modes with identical group velocities, leading to a local breakdown of dispersivity. This shows up as a geometric singularity of the resonant manifold and possibly as an infinite phase measure in the collision integral. Such singularities occur widely for classical wave systems, including acoustical waves, Rossby waves, helical waves in rotating fluids, light waves in nonlinear optics and also in quantum transport, e.g. kinetics of electron-hole excitations (matter waves) in graphene. These singularities are the exact analogue of the critical points found by Van Hove in 1953 for phonon dispersion relations in crystals. The importance of these singularities in wave kinetics depends on the dimension of phase space D = ( N − 2 ) d ( d physical space dimension, N the number of waves in resonance) and the degree of degeneracy δ of the critical points. Following Van Hove, we show that non-degenerate singularities lead to finite phase measures for D > 2 but produce divergences when D ≤ 2 and possible breakdown of wave kinetics if the collision integral itself becomes too large (or even infinite). Similar divergences and possible breakdown can occur for degenerate singularities, when D − δ ≤ 2 , as we find for several physical examples, including electron-hole kinetics in graphene. When the standard kinetic equation breaks down, then one must develop a new singular wave kinetics. We discuss approaches from pioneering 1971 work of Newell & Aucoin on multi-scale perturbation theory for acoustic waves and field-theoretic methods based on exact Schwinger-Dyson integral equations for the wave dynamics.


      PubDate: 2016-06-15T08:40:59Z
       
  • Dynamics of curved fronts in systems with power-law memory
    • Abstract: Publication date: 1 August 2016
      Source:Physica D: Nonlinear Phenomena, Volumes 328–329
      Author(s): M. Abu Hamed, A.A. Nepomnyashchy
      The dynamics of a curved front in a plane between two stable phases with equal potentials is modeled via two-dimensional fractional in time partial differential equation. A closed equation governing a slow motion of a small-curvature front is derived and applied for two typical examples of the potential function. Approximate axisymmetric and non-axisymmetric solutions are obtained.


      PubDate: 2016-06-15T08:40:59Z
       
  • Breathers in a locally resonant granular chain with precompression
    • Abstract: Publication date: 15 September 2016
      Source:Physica D: Nonlinear Phenomena, Volume 331
      Author(s): Lifeng Liu, Guillaume James, Panayotis Kevrekidis, Anna Vainchtein
      We study a locally resonant granular material in the form of a precompressed Hertzian chain with linear internal resonators. Using an asymptotic reduction, we derive an effective nonlinear Schrödinger (NLS) modulation equation. This, in turn, leads us to provide analytical evidence, subsequently corroborated numerically, for the existence of two distinct types of discrete breathers related to acoustic or optical modes: (a) traveling bright breathers with a strain profile exponentially vanishing at infinity and (b) stationary and traveling dark breathers, exponentially localized, time-periodic states mounted on top of a non-vanishing background. The stability and bifurcation structure of numerically computed exact stationary dark breathers is also examined. Stationary bright breathers cannot be identified using the NLS equation, which is defocusing at the upper edges of the phonon bands and becomes linear at the lower edge of the optical band.


      PubDate: 2016-06-15T08:40:59Z
       
  • Chevron folding patterns and heteroclinic orbits
    • Abstract: Publication date: Available online 11 May 2016
      Source:Physica D: Nonlinear Phenomena
      Author(s): Christopher J. Budd, Amine N. Chakhchoukh, Timothy J. Dodwell, Rachel Kuske
      We present a model of multilayer folding in which layers with bending stiffness E I are separated by a very stiff elastic medium of elasticity k 2 and subject to a horizontal load P . By using a dynamical systems analysis of the resulting fourth order equation, we show that as the end shortening per unit length E is increased, then if k 2 is large there is a smooth transition from small amplitude sinusoidal solutions at moderate values of P to larger amplitude chevron folds, with straight limbs separated by regions of high curvature when P is large. The chevron solutions take the form of near heteroclinic connections in the phase-plane. By means of this analysis, values for P and the slope of the limbs are calculated in terms of E and k 2 .


      PubDate: 2016-05-16T19:32:17Z
       
  • Stabilisation of difference equations with noisy prediction-based control
    • Abstract: Publication date: 1 July 2016
      Source:Physica D: Nonlinear Phenomena, Volume 326
      Author(s): E. Braverman, C. Kelly, A. Rodkina
      We consider the influence of stochastic perturbations on stability of a unique positive equilibrium of a difference equation subject to prediction-based control. These perturbations may be multiplicative x n + 1 = f ( x n ) − ( α + l ξ n + 1 ) ( f ( x n ) − x n ) , n = 0 , 1 , … , if they arise from stochastic variation of the control parameter, or additive x n + 1 = f ( x n ) − α ( f ( x n ) − x n ) + l ξ n + 1 , n = 0 , 1 , … , if they reflect the presence of systemic noise. We begin by relaxing the control parameter in the deterministic equation, and deriving a range of values for the parameter over which all solutions eventually enter an invariant interval. Then, by allowing the variation to be stochastic, we derive sufficient conditions (less restrictive than known ones for the unperturbed equation) under which the positive equilibrium will be globally a.s. asymptotically stable: i.e. the presence of noise improves the known effectiveness of prediction-based control. Finally, we show that systemic noise has a “blurring” effect on the positive equilibrium, which can be made arbitrarily small by controlling the noise intensity. Numerical examples illustrate our results.


      PubDate: 2016-05-16T19:32:17Z
       
  • A conservation law model for bidensity suspensions on an incline
    • Abstract: Publication date: Available online 11 May 2016
      Source:Physica D: Nonlinear Phenomena
      Author(s): Jeffrey T. Wong, Andrea L. Bertozzi
      We study bidensity suspensions of a viscous fluid on an incline. The particles migrate within the fluid due to a combination of gravity-induced settling and shear induced migration. We propose an extension a recent model (Murisic et al., 2013) for monodisperse suspensions to two species of particles, resulting in a hyperbolic system of three conservation laws for the height and particle concentrations. We analyze the Riemann problem and show that the system exhibits three-shock solutions representing distinct fronts of particles and liquid traveling at different speeds as well as singular shock solutions for sufficiently large concentrations, for which the mechanism is essentially the same as the single-species case. We also consider initial conditions describing a fixed volume of fluid, where solutions are rarefaction-shock pairs, and present a comparison to recent experimental results. The long-time behavior of solutions is identified for settled mono- and bidisperse suspensions and some leading-order asymptotics are derived in the single-species case for moderate concentrations.


      PubDate: 2016-05-16T19:32:17Z
       
  • Mechanics and polarity in cell motility
    • Abstract: Publication date: Available online 12 May 2016
      Source:Physica D: Nonlinear Phenomena
      Author(s): D. Ambrosi, A. Zanzottera
      The motility of a fish keratocyte on a flat substrate exhibits two distinct regimes: the non-migrating and the migrating one. In both configurations the shape is fixed in time and, when the cell is moving, the velocity is constant in magnitude and direction. Transition from a stable configuration to the other one can be produced by a mechanical or chemotactic perturbation. In order to point out the mechanical nature of such a bistable behaviour, we focus on the actin dynamics inside the cell using a minimal mathematical model. While the protein diffusion, recruitment and segregation govern the polarization process, we show that the free actin mass balance, driven by diffusion, and the polymerized actin retrograde flow, regulated by the active stress, are sufficient ingredients to account for the motile bistability. The length and velocity of the cell are predicted on the basis of the parameters of the substrate and of the cell itself. The key physical ingredient of the theory is the exchange among actin phases at the edges of the cell, that plays a central role both in kinematics and in dynamics.


      PubDate: 2016-05-16T19:32:17Z
       
  • Multi-bump solutions in a neural field model with external inputs
    • Abstract: Publication date: 1 July 2016
      Source:Physica D: Nonlinear Phenomena, Volume 326
      Author(s): Flora Ferreira, Wolfram Erlhagen, Estela Bicho
      We study the conditions for the formation of multiple regions of high activity or “bumps” in a one-dimensional, homogeneous neural field with localized inputs. Stable multi-bump solutions of the integro-differential equation have been proposed as a model of a neural population representation of remembered external stimuli. We apply a class of oscillatory coupling functions and first derive criteria to the input width and distance, which relate to the synaptic couplings that guarantee the existence and stability of one and two regions of high activity. These input-induced patterns are attracted by the corresponding stable one-bump and two-bump solutions when the input is removed. We then extend our analytical and numerical investigation to N -bump solutions showing that the constraints on the input shape derived for the two-bump case can be exploited to generate a memory of N > 2 localized inputs. We discuss the pattern formation process when either the conditions on the input shape are violated or when the spatial ranges of the excitatory and inhibitory connections are changed. An important aspect for applications is that the theoretical findings allow us to determine for a given coupling function the maximum number of localized inputs that can be stored in a given finite interval.


      PubDate: 2016-05-16T19:32:17Z
       
  • Existence and stability of PT-symmetric states in nonlinear
           two-dimensional square lattices
    • Abstract: Publication date: 1 July 2016
      Source:Physica D: Nonlinear Phenomena, Volume 326
      Author(s): Haitao Xu, P.G. Kevrekidis, Dmitry E. Pelinovsky
      Solitons and vortices symmetric with respect to simultaneous parity ( P ) and time reversing ( T ) transformations are considered on the square lattice in the framework of the discrete nonlinear Schrödinger equation. The existence and stability of such PT -symmetric configurations is analyzed in the limit of weak coupling between the lattice sites, when predictions on the elementary cell of a square lattice (i.e., a single square) can be extended to a large (yet finite) array of lattice cells. In particular, we find all examined vortex configurations are unstable with respect to small perturbations while a branch extending soliton configurations is spectrally stable. Our analytical predictions are found to be in good agreement with numerical computations.


      PubDate: 2016-05-16T19:32:17Z
       
  • Recurrence plots of discrete-time Gaussian stochastic processes
    • Abstract: Publication date: Available online 9 May 2016
      Source:Physica D: Nonlinear Phenomena
      Author(s): Sofiane Ramdani, Frédéric Bouchara, Julien Lagarde, Annick Lesne
      We investigate the statistical properties of recurrence plots (RPs) of data generated by discrete-time stationary Gaussian random processes. We analytically derive the theoretical values of the probabilities of occurrence of recurrence points and consecutive recurrence points forming diagonals on the RP, with an embedding dimension equal to 1 . These results allow us to obtain theoretical values of three measures: (i) the recurrence rate ( R E C ) (ii) the percent determinism ( D E T ) and (iii) RP-based estimation of the ε -entropy κ ( ε ) in the sense of correlation entropy. We apply these results to two Gaussian processes, namely the first order autoregressive process and fractional Gaussian noise. For these processes, we simulate a number of realizations and compare the RP-based estimations of the three selected measures to their theoretical values. These comparisons provide useful information on the quality of the estimations, such as the minimum required data length and threshold radius used to construct the RP.


      PubDate: 2016-05-10T19:21:51Z
       
  • Modulational instability and localized breather modes in the discrete
           nonlinear Schrödinger equation with helicoidal hopping
    • Abstract: Publication date: Available online 4 May 2016
      Source:Physica D: Nonlinear Phenomena
      Author(s): J. Stockhofe, P. Schmelcher
      We study a one-dimensional discrete nonlinear Schrödinger model with hopping to the first and a selected N th neighbor, motivated by a helicoidal arrangement of lattice sites. We provide a detailed analysis of the modulational instability properties of this equation, identifying distinctive multi-stage instability cascades due to the helicoidal hopping term. Bistability is a characteristic feature of the intrinsically localized breather modes, and it is shown that information on the stability properties of weakly localized solutions can be inferred from the plane-wave modulational instability results. Based on this argument, we derive analytical estimates of the critical parameters at which the fundamental on-site breather branch of solutions turns unstable. In the limit of large N , these estimates predict the emergence of an effective threshold behavior, which can be viewed as the result of a dimensional crossover to a two-dimensional square lattice.


      PubDate: 2016-05-05T19:00:55Z
       
  • Wavelet shrinkage of a noisy dynamical system with non-linear noise impact
    • Abstract: Publication date: 15 June 2016
      Source:Physica D: Nonlinear Phenomena, Volume 325
      Author(s): Matthieu Garcin, Dominique Guégan
      By filtering wavelet coefficients, it is possible to construct a good estimate of a pure signal from noisy data. Especially, for a simple linear noise influence, Donoho and Johnstone (1994) have already defined an optimal filter design in the sense of a minimization of the error made when estimating the pure signal. We set here a different framework where the influence of the noise is non-linear. In particular, we propose a method to filter the wavelet coefficients of a discrete dynamical system disrupted by a weak noise, in order to construct good estimates of the pure signal, including Bayes’ estimate, minimax estimate, oracular estimate or thresholding estimate. We present the example of a logistic and a Lorenz chaotic dynamical system as well as an adaptation of our technique in order to show empirically the robustness of the thresholding method in presence of leptokurtic noise. Moreover, we test both the hard and the soft thresholding and also another kind of smoother thresholding which seems to have almost the same reconstruction power as the hard thresholding. Finally, besides the tests on an estimated dataset, the method is tested on financial data: oil prices and NOK/USD exchange rate.


      PubDate: 2016-05-05T19:00:55Z
       
  • First-order aggregation models with alignment
    • Abstract: Publication date: 15 June 2016
      Source:Physica D: Nonlinear Phenomena, Volume 325
      Author(s): Razvan C. Fetecau, Weiran Sun, Changhui Tan
      We include alignment interactions in a well-studied first-order attractive–repulsive macroscopic model for aggregation. The distinctive feature of the extended model is that the equation that specifies the velocity in terms of the population density, becomes implicit, and can have non-unique solutions. We investigate the well-posedness of the model and show rigorously how it can be obtained as a macroscopic limit of a second-order kinetic equation. We work within the space of probability measures with compact support and use mass transportation ideas and the characteristic method as essential tools in the analysis. A discretization procedure that parallels the analysis is formulated and implemented numerically in one and two dimensions.


      PubDate: 2016-05-05T19:00:55Z
       
  • On the evolution of scattering data under perturbations of the Toda
           lattice
    • Abstract: Publication date: Available online 25 April 2016
      Source:Physica D: Nonlinear Phenomena
      Author(s): D. Bilman, I. Nenciu
      We present the results of an analytical and numerical study of the long-time behavior for certain Fermi-Pasta-Ulam (FPU) lattices viewed as perturbations of the completely integrable Toda lattice. Our main tools are the direct and inverse scattering transforms for doubly-infinite Jacobi matrices, which are well-known to linearize the Toda flow. We focus in particular on the evolution of the associated scattering data under the perturbed vs. the unperturbed equations. We find that the eigenvalues present initially in the scattering data converge to new, slightly perturbed eigenvalues under the perturbed dynamics of the lattice equation. To these eigenvalues correspond solitary waves that emerge from the solitons in the initial data. We also find that new eigenvalues emerge from the continuous spectrum as the lattice system is let to evolve under the perturbed dynamics.


      PubDate: 2016-04-29T18:42:49Z
       
  • Absolute stability and synchronization in neural field models with
           transmission delays
    • Abstract: Publication date: Available online 26 April 2016
      Source:Physica D: Nonlinear Phenomena
      Author(s): Chiu-Yen Kao, Chih-Wen Shih, Chang-Hong Wu
      Neural fields model macroscopic parts of the cortex which involve several populations of neurons. We consider a class of neural field models which are represented by integro-differential equations with transmission time delays which are space-dependent. The considered domains underlying the systems can be bounded or unbounded. A new approach, called sequential contracting, instead of the conventional Lyapunov functional technique, is employed to investigate the global dynamics of such systems. Sufficient conditions for the absolute stability and synchronization of the systems are established. Several numerical examples are presented to demonstrate the theoretical results.


      PubDate: 2016-04-29T18:42:49Z
       
  • Correlation functions of the KdV hierarchy and applications to
           intersection numbers over M¯g,n
    • Abstract: Publication date: Available online 29 April 2016
      Source:Physica D: Nonlinear Phenomena
      Author(s): Marco Bertola, Boris Dubrovin, Di Yang
      We derive an explicit generating function of correlations functions of an arbitrary tau-function of the KdV hierarchy. In particular applications, our formulation gives closed formulæ of a new type for the generating series of intersection numbers of ψ -classes as well as of mixed ψ - and κ -classes in full genera.


      PubDate: 2016-04-29T18:42:49Z
       
  • Nonlinear Dynamics on Interconnected Networks
    • Abstract: Publication date: 1 June 2016
      Source:Physica D: Nonlinear Phenomena, Volumes 323–324
      Author(s): Alex Arenas, Manlio De Domenico



      PubDate: 2016-04-29T18:42:49Z
       
  • On degree–degree correlations in multilayer networks
    • Abstract: Publication date: 1 June 2016
      Source:Physica D: Nonlinear Phenomena, Volumes 323–324
      Author(s): Guilherme Ferraz de Arruda, Emanuele Cozzo, Yamir Moreno, Francisco A. Rodrigues
      We propose a generalization of the concept of assortativity based on the tensorial representation of multilayer networks, covering the definitions given in terms of Pearson and Spearman coefficients. Our approach can also be applied to weighted networks and provides information about correlations considering pairs of layers. By analyzing the multilayer representation of the airport transportation network, we show that contrasting results are obtained when the layers are analyzed independently or as an interconnected system. Finally, we study the impact of the level of assortativity and heterogeneity between layers on the spreading of diseases. Our results highlight the need of studying degree–degree correlations on multilayer systems, instead of on aggregated networks.


      PubDate: 2016-04-29T18:42:49Z
       
  • Network bipartivity and the transportation efficiency of European
           passenger airlines
    • Abstract: Publication date: 1 June 2016
      Source:Physica D: Nonlinear Phenomena, Volumes 323–324
      Author(s): Ernesto Estrada, Jesús Gómez-Gardeñes
      The analysis of the structural organization of the interaction network of a complex system is central to understand its functioning. Here, we focus on the analysis of the bipartivity of graphs. We first introduce a mathematical approach to quantify bipartivity and show its implementation in general and random graphs. Then, we tackle the analysis of the transportation networks of European airlines from the point of view of their bipartivity and observe significant differences between traditional and low cost carriers. Bipartivity shows also that alliances and major mergers of traditional airlines provide a way to reduce bipartivity which, in its turn, is closely related to an increase of the transportation efficiency.


      PubDate: 2016-04-29T18:42:49Z
       
  • Asymptotic periodicity in networks of degrade-and-fire oscillators
    • Abstract: Publication date: 1 June 2016
      Source:Physica D: Nonlinear Phenomena, Volumes 323–324
      Author(s): Alex Blumenthal, Bastien Fernandez
      Networks of coupled degrade-and-fire (DF) oscillators are simple dynamical models of assemblies of interacting self-repressing genes. For mean-field interactions, which most mathematical studies have assumed so far, every trajectory must approach a periodic orbit. Moreover, asymptotic cluster distributions can be computed explicitly in terms of coupling intensity, and a massive collection of distributions collapses when this intensity passes a threshold. Here, we show that most of these dynamical features persist for an arbitrary coupling topology. In particular, we prove that, in any system of DF oscillators for which in and out coupling weights balance, trajectories with reasonable firing sequences must be asymptotically periodic, and periodic orbits are uniquely determined by their firing sequence. In addition to these structural results, illustrative examples are presented, for which the dynamics can be entirely described.


      PubDate: 2016-04-29T18:42:49Z
       
  • Cascades in interdependent flow networks
    • Abstract: Publication date: 1 June 2016
      Source:Physica D: Nonlinear Phenomena, Volumes 323–324
      Author(s): Antonio Scala, Pier Giorgio De Sanctis Lucentini, Guido Caldarelli, Gregorio D’Agostino
      In this manuscript, we investigate the abrupt breakdown behavior of coupled distribution grids under load growth. This scenario mimics the ever-increasing customer demand and the foreseen introduction of energy hubs interconnecting the different energy vectors. We extend an analytical model of cascading behavior due to line overloads to the case of interdependent networks and find evidence of first order transitions due to the long-range nature of the flows. Our results indicate that the foreseen increase in the couplings between the grids has two competing effects: on the one hand, it increases the safety region where grids can operate without withstanding systemic failures; on the other hand, it increases the possibility of a joint systems’ failure.
      Graphical abstract image

      PubDate: 2016-04-29T18:42:49Z
       
  • Erosion of synchronization: Coupling heterogeneity and network structure
    • Abstract: Publication date: 1 June 2016
      Source:Physica D: Nonlinear Phenomena, Volumes 323–324
      Author(s): Per Sebastian Skardal, Dane Taylor, Jie Sun, Alex Arenas
      We study the dynamics of network-coupled phase oscillators in the presence of coupling frustration. It was recently demonstrated that in heterogeneous network topologies, the presence of coupling frustration causes perfect phase synchronization to become unattainable even in the limit of infinite coupling strength. Here, we consider the important case of heterogeneous coupling functions and extend previous results by deriving analytical predictions for the total erosion of synchronization. Our analytical results are given in terms of basic quantities related to the network structure and coupling frustration. In addition to fully heterogeneous coupling, where each individual interaction is allowed to be distinct, we also consider partially heterogeneous coupling and homogeneous coupling in which the coupling functions are either unique to each oscillator or identical for all network interactions, respectively. We demonstrate the validity of our theory with numerical simulations of multiple network models, and highlight the interesting effects that various coupling choices and network models have on the total erosion of synchronization. Finally, we consider some special network structures with well-known spectral properties, which allows us to derive further analytical results.


      PubDate: 2016-04-29T18:42:49Z
       
  • Contact-based model for strategy updating and evolution of cooperation
    • Abstract: Publication date: 1 June 2016
      Source:Physica D: Nonlinear Phenomena, Volumes 323–324
      Author(s): Jianlei Zhang, Zengqiang Chen
      To establish an available model for the astoundingly strategy decision process of players is not easy, sparking heated debate about the related strategy updating rules is intriguing. Models for evolutionary games have traditionally assumed that players imitate their successful partners by the comparison of respective payoffs, raising the question of what happens if the game information is not easily available. Focusing on this yet-unsolved case, the motivation behind the work presented here is to establish a novel model for the updating of states in a spatial population, by detouring the required payoffs in previous models and considering much more players’ contact patterns. It can be handy and understandable to employ switching probabilities for determining the microscopic dynamics of strategy evolution. Our results illuminate the conditions under which the steady coexistence of competing strategies is possible. These findings reveal that the evolutionary fate of the coexisting strategies can be calculated analytically, and provide novel hints for the resolution of cooperative dilemmas in a competitive context. We hope that our results have disclosed new explanations about the survival and coexistence of competing strategies in structured populations.


      PubDate: 2016-04-29T18:42:49Z
       
  • Consensus dynamics on random rectangular graphs
    • Abstract: Publication date: 1 June 2016
      Source:Physica D: Nonlinear Phenomena, Volumes 323–324
      Author(s): Ernesto Estrada, Matthew Sheerin
      A random rectangular graph (RRG) is a generalization of the random geometric graph (RGG) in which the nodes are embedded into a rectangle with side lengths a and b = 1 / a , instead of on a unit square [ 0 , 1 ] 2 . Two nodes are then connected if and only if they are separated at a Euclidean distance smaller than or equal to a certain threshold radius r . When a = 1 the RRG is identical to the RGG. Here we apply the consensus dynamics model to the RRG. Our main result is a lower bound for the time of consensus, i.e., the time at which the network reaches a global consensus state. To prove this result we need first to find an upper bound for the algebraic connectivity of the RRG, i.e., the second smallest eigenvalue of the combinatorial Laplacian of the graph. This bound is based on a tight lower bound found for the graph diameter. Our results prove that as the rectangle in which the nodes are embedded becomes more elongated, the RRG becomes a ’large-world’, i.e., the diameter grows to infinity, and a poorly-connected graph, i.e., the algebraic connectivity decays to zero. The main consequence of these findings is the proof that the time of consensus in RRGs grows to infinity as the rectangle becomes more elongated. In closing, consensus dynamics in RRGs strongly depend on the geometric characteristics of the embedding space, and reaching the consensus state becomes more difficult as the rectangle is more elongated.


      PubDate: 2016-04-29T18:42:49Z
       
  • Interplay between consensus and coherence in a model of interacting
           opinions
    • Abstract: Publication date: 1 June 2016
      Source:Physica D: Nonlinear Phenomena, Volumes 323–324
      Author(s): Federico Battiston, Andrea Cairoli, Vincenzo Nicosia, Adrian Baule, Vito Latora
      The formation of agents’ opinions in a social system is the result of an intricate equilibrium among several driving forces. On the one hand, the social pressure exerted by peers favors the emergence of local consensus. On the other hand, the concurrent participation of agents to discussions on different topics induces each agent to develop a coherent set of opinions across all the topics in which he/she is active. Moreover, the pervasive action of external stimuli, such as mass media, pulls the entire population towards a specific configuration of opinions on different topics. Here we propose a model in which agents with interrelated opinions, interacting on several layers representing different topics, tend to spread their own ideas to their neighborhood, strive to maintain internal coherence, due to the fact that each agent identifies meaningful relationships among its opinions on the different topics, and are at the same time subject to external fields, resembling the pressure of mass media. We show that the presence of heterogeneity in the internal coupling assigned by agents to their different opinions allows to obtain states with mixed levels of consensus, still ensuring that all the agents attain a coherent set of opinions. Furthermore, we show that all the observed features of the model are preserved in the presence of thermal noise up to a critical temperature, after which global consensus is no longer attainable. This suggests the relevance of our results for real social systems, where noise is inevitably present in the form of information uncertainty and misunderstandings. The model also demonstrates how mass media can be effectively used to favor the propagation of a chosen set of opinions, thus polarizing the consensus of an entire population.


      PubDate: 2016-04-29T18:42:49Z
       
  • Quasi-steady state reduction for compartmental systems
    • Abstract: Publication date: Available online 21 April 2016
      Source:Physica D: Nonlinear Phenomena
      Author(s): Alexandra Goeke, Christian Lax
      We present a method to determine an asymptotic reduction (in the sense of Tikhonov and Fenichel) for singularly perturbed compartmental systems in the presence of slow transport. It turns out that the reduction can be derived from the individual interaction terms alone. We apply the result to spatially discretized reaction-diffusion systems and obtain (based on the reduced discretized systems) a heuristic to reduce reaction-diffusion systems in presence of slow diffusion.


      PubDate: 2016-04-24T18:22:14Z
       
  • The extended Estabrook-Wahlquist method
    • Abstract: Publication date: Available online 21 April 2016
      Source:Physica D: Nonlinear Phenomena
      Author(s): S. Roy Choudhury, Matthew Russo
      Variable Coefficient Korteweg de Vries (vcKdV), modified Korteweg de Vries (vcMKdV), and nonlinear Schröedinger (NLS) equations have a long history dating from their derivation in various applications. A technique based on extended Lax Pairs has been devised recently to derive time-and-space-dependent-coefficient generalizations of various such Lax-integrable NLPDE hierarchies, which are thus more general than almost all cases considered earlier via methods such as the Painlevé Test, Bell Polynomials, and similarity methods. However, this technique, although operationally effective, has the significant disadvantage that, for any integrable system with spatiotemporally varying coefficients, one must ‘guess’ a generalization of the structure of the known Lax Pair for the corresponding system with constant coefficients. Motivated by the somewhat arbitrary nature of the above procedure, we embark in this paper on an attempt to systematize the derivation of Lax-integrable systems with variable coefficients. We consider the Estabrook-Wahlquist (EW) prolongation technique, a relatively self-consistent procedure requiring little prior information. However, this immediately requires that the technique be significantly generalized in several ways, including solving matrix partial differential equations instead of algebraic ones as the structure of the Lax Pair is systematically computed, and also in solving the constraint equations to deduce the explicit forms for various ‘coefficient’ matrices. The new and extended EW technique which results is illustrated by algorithmically deriving generalized Lax-integrable versions of the NLS, generalized fifth-order KdV, MKdV, and derivative nonlinear Schröedinger (DNLS) equations. We also show how this method correctly excludes the existence of a nontrivial Lax pair for a nonintegrable NLPDE such as the variable-coefficient cubic-quintic NLS.


      PubDate: 2016-04-24T18:22:14Z
       
  • Cellular replication limits in the Luria-Delbrück mutation model
    • Abstract: Publication date: Available online 19 April 2016
      Source:Physica D: Nonlinear Phenomena
      Author(s): Ignacio A. Rodriguez-Brenes, Dominik Wodarz, Natalia L. Komarova
      Originally developed to elucidate the mechanisms of natural selection in bacteria, the Luria-Delbrück model assumed that cells are intrinsically capable of dividing an unlimited number of times. This assumption however, is not true for human somatic cells which undergo replicative senescence. Replicative senescence is thought to act as a mechanism to protect against cancer and the escape from it is a rate-limiting step in cancer progression. Here we introduce a Luria-Delbrück model that explicitly takes into account cellular replication limits in the wild type cell population and models the emergence of mutants that escape replicative senescence. We present results on the mean, variance, distribution, and asymptotic behavior of the mutant population in terms of three classical formulations of the problem. More broadly the paper introduces the concept of incorporating replicative limits as part of the Luria-Delbrück mutational framework. Guidelines to extend the theory to include other types of mutations and possible applications to the modeling of telomere crisis and fluctuation analysis are also discussed.


      PubDate: 2016-04-19T18:05:31Z
       
  • A trajectory-free framework for analysing multiscale systems
    • Abstract: Publication date: Available online 19 April 2016
      Source:Physica D: Nonlinear Phenomena
      Author(s): Gary Froyland, Georg A. Gottwald, Andy Hammerlindl
      We develop algorithms built around properties of the transfer operator and Koopman operator which 1) test for possible multiscale dynamics in a given dynamical system, 2) estimate the magnitude of the time-scale separation, and finally 3) distill the reduced slow dynamics on a suitably designed subspace. By avoiding trajectory integration, the developed techniques are highly computationally efficient. We corroborate our findings with numerical simulations of a test problem.


      PubDate: 2016-04-19T18:05:31Z
       
  • Multi-soliton, multi-breather and higher order rogue wave solutions to the
           complex short pulse equation
    • Abstract: Publication date: Available online 19 April 2016
      Source:Physica D: Nonlinear Phenomena
      Author(s): Liming Ling, Bao-Feng Feng, Zuonong Zhu
      In the present paper, we are concerned with the general analytic solutions to the complex short pulse (CSP) equation including soliton, breather and rogue wave solutions. With the aid of a generalized Darboux transformation, we construct the N -bright soliton solution in a compact determinant form, the N -breather solution including the Akhmediev breather and a general higher order rogue wave solution. The first and second order rogue wave solutions are given explicitly and analyzed. The asymptotic analysis is performed rigourously for both the N -soliton and the N -breather solutions. All three forms of the analytical solutions admit either smoothed-, cusped- or looped-type ones for the CSP equation depending on the parameters. It is noted that, due to the reciprocal (hodograph) transformation, the rogue wave solution to the CSP equation can be a smoothed, cusponed or a looped one, which is different from the rogue wave solution found so far.


      PubDate: 2016-04-19T18:05:31Z
       
  • Nonlinear optical vibrations of single-walled carbon nanotubes. 1. Energy
           exchange and localization of low-frequency oscillations
    • Abstract: Publication date: Available online 4 April 2016
      Source:Physica D: Nonlinear Phenomena
      Author(s): V.V. Smirnov, L.I. Manevitch, M. Strozzi, F. Pellicano
      We present the results of analytical study and molecular dynamics simulation of low energy nonlinear non-stationary dynamics of single-walled carbon nanotubes (CNTs). New phenomena of intense energy exchange between different parts of CNT and weak energy localization in the excited part of CNT are analytically predicted in the framework of the continuum shell theory. Their origin is clarified by means of the concept of Limiting Phase Trajectory, and the analytical results are confirmed by the molecular dynamics simulation of simply supported CNTs.


      PubDate: 2016-04-04T17:26:25Z
       
  • Inertial effects on thin-film wave structures with imposed surface shear
           on an inclined plane
    • Abstract: Publication date: Available online 21 March 2016
      Source:Physica D: Nonlinear Phenomena
      Author(s): M. Sivapuratharasu, S. Hibberd, M.E. Hubbard, H. Power
      This study provides an extended approach to the mathematical simulation of thin-film flow on a flat inclined plane relevant to flows subject to high surface shear. Motivated by modelling thin-film structures within an industrial context, wave structures are investigated for flows with moderate inertial effects and small film depth aspect ratio ε . Approximations are made assuming a Reynolds number, Re ∼ O ( ε − 1 ) and depth-averaging used to simplify the governing Navier-Stokes equations. A parallel Stokes flow is expected in the absence of any wave disturbance and a generalisation for the flow is based on a local quadratic profile. This approch provides a more general system which includes inertial effects and is solved numerically. Flow structures are compared with studies for Stokes flow in the limit of negligible inertial effects. Both two-tier and three-tier wave disturbances are used to study film profile evolution. A parametric study is provided for wave disturbances with increasing film Reynolds number. An evaluation of standing wave and transient film profiles is undertaken and identifies new profiles not previously predicted when inertial effects are neglected.


      PubDate: 2016-03-25T13:22:38Z
       
  • Modulational instability in nonlinear nonlocal equations of regularized
           long wave type
    • Abstract: Publication date: Available online 18 March 2016
      Source:Physica D: Nonlinear Phenomena
      Author(s): Vera Mikyoung Hur, Ashish Kumar Pandey
      We study the stability and instability of periodic traveling waves in the vicinity of the origin in the spectral plane, for equations of Benjamin-Bona-Mahony (BBM) and regularized Boussinesq types permitting nonlocal dispersion. We extend recent results for equations of Korteweg-de Vries type and derive modulational instability indices as functions of the wave number of the underlying wave. We show that a sufficiently small, periodic traveling wave of the BBM equation is spectrally unstable to long wavelength perturbations if the wave number is greater than a critical value and a sufficiently small, periodic traveling wave of the regularized Boussinesq equation is stable to square integrable perturbations.


      PubDate: 2016-03-21T13:19:25Z
       
  • Jump bifurcations in some degenerate planar piecewise linear differential
           systems with three zones
    • Abstract: Publication date: Available online 11 March 2016
      Source:Physica D: Nonlinear Phenomena
      Author(s): Rodrigo Euzébio, Rubens Pazim, Enrique Ponce
      We consider continuous piecewise-linear differential systems with three zones where the central one is degenerate, that is, the determinant of its linear part vanishes. By moving one parameter which is associated to the equilibrium position, we detect some new bifurcations exhibiting jump transitions both in the equilibrium location and in the appearance of limit cycles. In particular, we introduce the scabbard bifurcation, characterized by the birth of a limit cycle from a continuum of equilibrium points. Some of the studied bifurcations are detected, after an appropriate choice of parameters, in a piecewise linear Morris-Lecar model for the activity of a single neuron activity, which is usually considered as a reduction of the celebrated Hodgkin-Huxley equations.


      PubDate: 2016-03-16T13:03:26Z
       
  • A numerical method for computing initial conditions of Lagrangian
           invariant tori using the frequency map
    • Abstract: Publication date: Available online 10 March 2016
      Source:Physica D: Nonlinear Phenomena
      Author(s): Alejandro Luque, Jordi Villanueva
      We present a numerical method for computing initial conditions of Lagrangian quasi-periodic invariant tori of Hamiltonian systems and symplectic maps. Such initial conditions are found by solving, using the Newton method, a nonlinear system obtained by imposing suitable conditions on the frequency map. The basic tool is a newly developed methodology to perform the frequency analysis of a discrete quasi-periodic signal, allowing to compute frequencies and their derivatives with respect to parameters. Roughly speaking, this method consists in computing suitable weighted averages of the iterates of the signal and using the Richardson extrapolation method. The proposed approach performs with high accuracy at a moderate computational cost. We illustrate the method by considering a discrete FPU model and the vicinity of the point L 4 in a RTBP.


      PubDate: 2016-03-11T12:45:40Z
       
  • Dressing method for the vector sine-Gordon equation and its soliton
           interactions
    • Abstract: Publication date: Available online 9 March 2016
      Source:Physica D: Nonlinear Phenomena
      Author(s): Alexander V. Mikhailov, Georgios Papamikos, Jing Ping Wang
      In this paper, we develop the dressing method to study the exact solutions for the vector sine-Gordon equation. The explicit formulas for one kink and one breather are derived. The method can be used to construct multi-soliton solutions. Two soliton interactions are also studied. The formulas for position shift of the kink and position and phase shifts of the breather are given. These quantities only depend on the pole positions of the dressing matrices.


      PubDate: 2016-03-11T12:45:40Z
       
  • Three–dimensional representations of the tube manifolds of the
           planar restricted three–body problem
    • Abstract: Publication date: Available online 10 March 2016
      Source:Physica D: Nonlinear Phenomena
      Author(s): Elena Lega, Massimiliano Guzzo
      The stable and unstable manifolds of the Lyapunov orbits of the Lagrangian equilibrium points L1, L2 play a key role in the understanding of the complicated dynamics of the circular restricted three–body problem. By developing a recent technique of computation of the stable and unstable manifolds, based on the use of Fast Lyapunov Indicators modified by the introduction of a filtering window function, we compute sample three–dimensional representations of the manifolds which show an original vista about their complicated development in the phase-space.


      PubDate: 2016-03-11T12:45:40Z
       
  • Hopf normal form with SN symmetry and reduction to systems of nonlinearly
           coupled phase oscillators
    • Abstract: Publication date: Available online 26 February 2016
      Source:Physica D: Nonlinear Phenomena
      Author(s): Peter Ashwin, Ana Rodrigues
      Coupled oscillator models where N oscillators are identical and symmetrically coupled to all others with full permutation symmetry S N are found in a variety of applications. Much, but not all, work on phase descriptions of such systems consider the special case of pairwise coupling between oscillators. In this paper, we show this is restrictive - and we characterise generic multi-way interactions between oscillators that are typically present, except at the very lowest order near a Hopf bifurcation where the oscillations emerge. We examine a network of identical weakly coupled dynamical systems that are close to a supercritical Hopf bifurcation by considering two parameters, ϵ (the strength of coupling) and λ (an unfolding parameter for the Hopf bifurcation). For small enough λ > 0 there is an attractor that is the product of N stable limit cycles; this persists as a normally hyperbolic invariant torus for sufficiently small ϵ > 0 . Using equivariant normal form theory, we derive a generic normal form for a system of coupled phase oscillators with S N symmetry. For fixed N and taking the limit 0 < ϵ ≪ λ ≪ 1 , we show that the attracting dynamics of the system on the torus can be well approximated by a coupled phase oscillator system that, to lowest order, is the well-known Kuramoto-Sakaguchi system of coupled oscillators. The next order of approximation genericlly includes terms with up to four interacting phases, regardless of N . Using a normalization that maintains nontrivial interactions in the limit N → ∞ , we show that the additional terms can lead to new phenomena in terms of coexistence of two-cluster states with the same phase difference but different cluster size.


      PubDate: 2016-03-07T12:36:02Z
       
  • Traveling wave solutions in a chain of periodically forced coupled
           nonlinear oscillators
    • Abstract: Publication date: Available online 27 February 2016
      Source:Physica D: Nonlinear Phenomena
      Author(s): M. Duanmu, N. Whitaker, P.G. Kevrekidis, A. Vainchtein, J.E. Rubin
      Motivated by earlier studies of artificial perceptions of light called phosphenes, we analyze traveling wave solutions in a chain of periodically forced coupled nonlinear oscillators modeling this phenomenon. We examine the discrete model problem in its co-traveling frame and systematically obtain the corresponding traveling waves in one spatial dimension. Direct numerical simulations as well as linear stability analysis are employed to reveal the parameter regions where the traveling waves are stable, and these waves are, in turn, connected to the standing waves analyzed in earlier work. We also consider a two-dimensional extension of the model and demonstrate the robust evolution and stability of planar fronts. Our simulations also suggest the radial fronts tend to either annihilate or expand and flatten out, depending on the phase value inside and the parameter regime. Finally, we observe that solutions that initially feature two symmetric fronts with bulged centers evolve in qualitative agreement with experimental observations of phosphenes.


      PubDate: 2016-03-07T12:36:02Z
       
  • Filter accuracy for the Lorenz 96 model: Fixed versus adaptive observation
           operators
    • Abstract: Publication date: Available online 23 February 2016
      Source:Physica D: Nonlinear Phenomena
      Author(s): K.J.H. Law, D. Sanz-Alonso, A. Shukla, A.M. Stuart
      In the context of filtering chaotic dynamical systems it is well-known that partial observations, if sufficiently informative, can be used to control the inherent uncertainty due to chaos. The purpose of this paper is to investigate, both theoretically and numerically, conditions on the observations of chaotic systems under which they can be accurately filtered. In particular, we highlight the advantage of adaptive observation operators over fixed ones. The Lorenz ’96 model is used to exemplify our findings. We consider discrete-time and continuous-time observations in our theoretical developments. We prove that, for fixed observation operator, the 3DVAR filter can recover the system state within a neighbourhood determined by the size of the observational noise. It is required that a sufficiently large proportion of the state vector is observed, and an explicit form for such sufficient fixed observation operator is given. Numerical experiments, where the data is incorporated by use of the 3DVAR and extended Kalman filters, suggest that less informative fixed operators than given by our theory can still lead to accurate signal reconstruction. Adaptive observation operators are then studied numerically; we show that, for carefully chosen adaptive observation operators, the proportion of the state vector that needs to be observed is drastically smaller than with a fixed observation operator. Indeed, we show that the number of state coordinates that need to be observed may even be significantly smaller than the total number of positive Lyapunov exponents of the underlying system.


      PubDate: 2016-02-24T12:04:11Z
       
 
 
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